Evaluate: tan-1(tan 5π/6) + cos-1(cos 13π/6)
Solution:
Step 1: Use principal value ranges
tan-1(x) ∈ \((-π/2, π/2)\) cos-1(x) ∈ \([0, π]\)
Step 2: Evaluate tan⁻¹(tan 5π/6)
\[ \tan \frac{5\pi}{6} = -\tan \frac{\pi}{6} = -\frac{1}{\sqrt{3}} \]
\[ \tan^{-1}\left(-\frac{1}{\sqrt{3}}\right) = -\frac{\pi}{6} \]
Step 3: Evaluate cos⁻¹(cos 13π/6)
\[ \cos \frac{13\pi}{6} = \cos\left(2\pi + \frac{\pi}{6}\right) = \cos \frac{\pi}{6} = \frac{\sqrt{3}}{2} \]
\[ \cos^{-1}\left(\frac{\sqrt{3}}{2}\right) = \frac{\pi}{6} \]
Step 4: Add values
\[ -\frac{\pi}{6} + \frac{\pi}{6} = 0 \]
Final Answer:
Value = \[ 0 \]